More weird and wonderful observations about the expansion of the universe

In high school physics we are taught the Galilean principle of relativity (named after Galileo Galilei, not the biblical land of Galilee). That is that there is no such thing as absolute velocity. All velocity is relative. Consequently we cannot ‘feel’ velocity. If we are inside a closed compartment with no windows, we cannot tell whether it is ‘moving’. We might think we can, and compare the feeling of being in a moving railway carriage or aeroplane to being in that same vehicle when it is ‘stopped’. However, as our high school science teachers explained to us, the feeling of motion we experience there is not the general forward (horizontal) motion but the minor vertical or horizontal accelerations that happen when the vehicle goes over bumps or gaps in the rails (train) or air (plane). If the rails or air were perfectly smooth, and we were travelling straight – not turning – we would not be able to tell whether we were ‘moving’.

If our science teachers were especially good, they might have summed this up for us as ‘you can feel acceleration, but you can’t feel motion’.

In my earlier essay ‘Expansion of the Universe’ I pointed out that it is quite possible for two items to be moving apart at a rate faster than the speed of light, and that that does not contradict Einstein’s theory of relativity, as long as those two items are sufficiently far apart. The ‘acceleration’ that arises from the cosmological constant (or ‘dark energy’, if you prefer spooky mystical terms) is so mild that we cannot not feel it.

It has since occurred to me though, that, because that rate of relative acceleration increases with the distance between two objects, there must be objects immensely far from us, relative to which our acceleration will be very large – greater than the several ‘g’s needed to make a person black out. This observation follows from Hubble’s equation.

Those that don’t like maths had better close their eyes for a few seconds. Don’t worry, it won’t last long.

Hubble’s equation tells us that dr/dt = H r, where r is the ‘comoving’ distance from us to a distant galaxy, dr/dt is the rate of change of that distance (using ‘cosmological time’ for t) and H is the Hubble parameter (2.2 x 10-21 s-1). The solution to this differential equation is r = r0 eHt. where r0 is the current distance from us to the distant galaxy. Hence the relative acceleration between us and the distant galaxy is d2r/dt2 = r0H2eHt = H2r.

Almost any human will lose consciousness under an acceleration (‘g force’) of 10 ‘g’s, (‘g’ is the acceleration due to gravity at the Earth’s surface). That is 98ms-2. To find the distance away of objects that are accelerating relative to us at that rate we set 98ms-2 = H2r. So r = 98ms-2/H2 = 98ms-2/ (2.2 x 10-21 s-1)2 = 2 x 1043m = 2 x 1018 billion light years. That is almost 5 x 1016 (50 quadrillion) times the radius of the observable universe.

You can open your eyes again now.

We have just worked out that the relative acceleration between ourselves and galaxies that are 2 x 1018 billion light years away is around 10 ‘g’s. One or both of us and the distant galaxy must be accelerating away from the other, and the sum of those two accelerations must be 10g. The Cosmological Principle tells us that the universe is essentially the same all over (‘homogeneity’ and ‘isotropy’), so we should have the same acceleration as that galaxy, which means we should each be accelerating at 5g, in opposite directions. That would be enough to pull the Earth out of its orbit around the Sun, and to pull us off the Earth’s surface. Also, we should be able to feel it – in fact it should make us feel very sick, like a super-extreme roller coaster. Remember that science teacher that told us you can’t feel motion but you can feel acceleration? So why can’t we feel it?

The answer is actually at the heart of Einstein’s General Theory of Relativity and, fortunately, it’s not highly technical. What we can feel is acceleration, but in this context it is an oversimplification to say that acceleration is the rate of change of our speed. In General Relativity the acceleration of a body means the rate of divergence of the body’s path through spacetime (its ‘worldline’, in relativistic jargon), from the tangent geodesic. Let’s unpack this. A geodesic is the equivalent of a straight line on a curvy surface. For instance the ‘great circles’ that are lines of longitude on the Earth are geodesics, and so is the Equator (but not lines of latitude like the two tropics or the Arctic and Antarctic circles). At any point in your life, the tangent geodesic to your body is the path it would travel through four-dimensional spacetime if there were no non-gravitational forces acting on it. Fortunately for you, there are non-gravitational forces acting on you (principally from the ground, pushing up on the soles of your feet), otherwise you would follow your tangent geodesic which is a headlong plummet towards the centre of the Earth. The floor is constantly pushing you away from your tangent geodesic in four dimensions, just as a turned steering wheel pushes a car away from the line of the beam of its headlights (which is the tangent in three-dimensional space to the car’s curved 3D path).

The acceleration you feel, which is just your feeling of weight, arises because your body, by not plummeting, is constantly diverging from its tangent geodesic. You might think you don’t feel any acceleration when you’re just standing still on the floor, or lying in bed, but that’s because you have felt it ever since you were first snuggled up in your mother’s womb, so you don’t notice it. What you do notice is if the acceleration suddenly stops. That’s what that ‘stomach in your mouth’ feeling is that you get when you jump off a pier into the sea and are briefly ‘weightless’. For a second or two you are following your tangent geodesic, and it feels odd because you’re not used to it.

The science teacher’s advice, that you can feel acceleration, needs to be refined to specify that you only feel ‘General Relativistic acceleration’, which is not rate of change of speed (as it is described in Year 11 Physics) but divergence from our tangent geodesic*. The two are only the same in non-accelerated (‘inertial’) frames of reference. In high school Physics we usually only deal with inertial frames of reference so the distinction doesn’t matter. But it does when we are considering our position relative to very distant galaxies.

Now how does this explain our inability to feel our acceleration relative to that distant galaxy? The answer is that the Earth is following its tangent geodesic which, as well as orbiting the Sun, is heading increasingly rapidly away from the distant galaxy. So we can’t feel that change in motion because it is not diverging from the tangent geodesic. The only thing we can feel is the divergence that arises from the floor pushing us up in opposition to the Earth’s gravity.

In summary, the only ‘acceleration’ you can feel is that which pushes you away from your tangent geodesic. It doesn’t matter how wildly that geodesic may be curving through spacetime – you will never be able to feel it.

* Note: Technically speaking, the strength of acceleration that you feel – the ’g force’ – is the magnitude of the Covariant Derivative of your four-velocity in its current direction.

Cosmic Inflation

A much more extreme example of this is in the ‘inflationary era’ of the universe, which is believed to have occupied the first unimaginably tiny fraction of a second after the Big Bang (10-32 seconds). In that sliver of time every part of the universe is thought to have expanded by a linear factor of around 1026. Imagine if you had just put your pencil down for an instant, then looked to find it and noticed that it was now a billion light years away. Wouldn’t that be annoying? Fortunately, there were no humans around at the time to be annoyed by such nuisances, and not even any pencils either. I do sometimes wonder though whether a brief, local, reappearance of cosmological inflation might be the reason for the occasional mysterious disappearance of my socks.

Describing that period of inflation in terms of acceleration, the g forces are mind-boggling. If we take a brutally crude and almost certainly horribly wrong (but reasonable enough to make the point) assumption that the acceleration was constant, we can use the high school formula to calculate the average acceleration during this split second.

Mathophobes look away again briefly:

The formula is distance (r) = ½ a t2 where a is the acceleration, t is the time period and r is how far away the object is at the end of the period. This gives a = 2r / t2. If I put my pencil down one metre away and 10-32 seconds it is 1026 metres away we have a = 2 x 1026m / (10-32s)2 = 2 x 1090ms-2.

The average acceleration during the inflationary split-second was 2 x 1089 ‘g’s! That would give you a bit of a tummy ache.

Only it wouldn’t, for the same reason as why we can’t feel our acceleration relative to the distant galaxy. We would just be following our tangent geodesic, and although that geodesic was horrendously curved, we wouldn’t feel a thing, because we only feel divergence from that path.

Only it would be rather uncomfortable because our bodies take up space, and so have many different geodesics going through them. Those geodesics, although initially very close, would diverge from one another so rapidly that a moment later they would be separated by squillions of light years. So your body would be ripped apart by the expansion of space, and the molecules that were neatly collected together to make your kidney one instant would be spread over an unimaginably vast expanse of space an instant later. Fortunately, it would happen so fast that you’d never notice. Even more fortunately, we can be fairly sure that nobody reading this was born before 1879, and hence they probably weren’t around to experience such a cosmic evisceration.

Another odd thing is that none of the molecules in your kidney would break the cosmic speed limit of c (the speed of light), even though they become separated by many light years in a tiny fraction of a second. This relies on the (somewhat technical) fact that the molecules not only follow their geodesics, but that those geodesics are timelike. It is only non-timelike geodesics that are forbidden. The concept of timelike things is partially explained in my essay ‘Expansion of the Universe’ . For a full explanation, see any good relativity textbook such as Bernard Schutz’s ‘A First Course in General Relativity’.

Summary

We can only feel acceleration, not constant motion. But the acceleration we can feel is only ‘General Relativistic acceleration’ that is divergence of our path from our tangent geodesic. Acceleration that is just an increasing rate of separation between us and something else will not be felt if it is solely because of our following geodesics of spacetime. Such non-General Relativistic Acceleration exists between us and immensely distant galaxies. For sufficiently distant galaxies, that acceleration is greater than that of a ferocious roller-coaster, yet we cannot feel it at all.

In the first split second of our universe, non-General Relativistic acceleration is believed to have occurred – called ‘cosmic inflation’ – that is unimaginably huge in terms of ‘g forces’. The acceleration could not have been felt if we had been there at the time, but it would have instantly ripped apart any object that was there, because of the extreme divergence of temporarily nearby geodesics.